Finite Group Databasebeta

← #70282

Subgroups — #70282

Order 384.

406 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 93 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#944C211yesmaximal, characteristic, complemented
2C22×#215C211yesmaximal, characteristic
3C22×#204C211yesmaximal, characteristic
4C2×C3×#103C211yesmaximal, characteristic, complemented
5#4988C211yesmaximal, characteristic
6C2×#810C211yesmaximal, characteristic, complemented
7#4550C211yesmaximal, characteristic, complemented
8#2013—31nomaximal, Sylow
9C22×C3×Q8C2211yescharacteristic
10C22×D12C2211yescharacteristic
11C22×Dic6C2211yescharacteristic
12C22×C3×C8C2211yescharacteristic
13C2×C3×C4⋊C4C2211yescharacteristic
14#944—22no—
15#944—22no—
16C2×#215—22no—
17C2×#215—22no—
18C2×#215—22no—
19C2×(C2×D3)⋊C4—22no—
20C2×Dic3⋊C4—22no—
21C2×C4×Dic3C2211yescharacteristic
22C22×C3⋊C8C2211yescharacteristic
23C2×#204—22no—
24C2×#204—22no—
25C2×#204—22no—
26C3×#103—22no—
27#810—22no—
28C22×SD16—31no—
29C22×SD16—31no—
30C2×C4⋊Q8—31no—
31C2×#103—31no—
32C2×#103D311yescharacteristic, complemented
33#390—31no—
34#336—31no—
35C23×D3—21no—
36C2×C3×Q8—21no—
37C2×C3×Q8—21no—
38C2×C3×Q8—21no—
39C2×C3×Q8—21no—
40C2×C3×Q8—21no—
41C2×C3×Q8D412yes—
42C22×C3×C4—21no—
43C22×C3×C4—21no—
44C22×C3×C4C2311yescharacteristic
45C22×Dic3—21no—
46C22×Dic3D412yes—
47C2×D12—21no—
48C2×D12—21no—
49C2×D12—21no—
50C2×D12—21no—
51C2×D12—21no—
52C2×D12D412yes—
53C2×Dic6—21no—
54C2×Dic6—21no—
55C2×Dic6—21no—
56C2×Dic6—21no—
57C2×Dic6—21no—
58C2×Dic6D412yes—
59C2×C3×C8—21no—
60C2×C3×C8—21no—
61C2×C3×C8D412yes—
62C3×C4⋊C4—21no—
63C3×C4⋊C4—21no—
64#215—28no—
65(C2×D3)⋊C4—44no—
66Dic3⋊C4—28no—
67C4×Dic3—21no—
68C4×Dic3—21no—
69C2×C3⋊C8—21no—
70C2×C3⋊C8—21no—
71C2×C3⋊C8D412yes—
72#204—28no—
73C22×Q8—31no—
74C22×Q8C2×D311yescharacteristic
75C22×D4—31no—
76C2×SD16—62no—
77C2×SD16—62no—
78C2×SD16—62no—
79C2×SD16—62no—
80C2×SD16—62no—
81C2×SD16—62no—
82C22×C8—31no—
83C22×C8C2×D311yescharacteristic
84C4⋊Q8—62no—
85C4⋊Q8—62no—
86C2×C4⋊C4—62no—
87C2×C4⋊C4C2×D311yescharacteristic
88C2×C22⋊C4—62no—
89C2×C42—31no—
90#103—22no—
91#103—62no—
92C23×C3C2×D411yescharacteristic
93C22×D3—22no—
94C22×D3—41no—
95C22×D3—41no—
96C22×D3—42no—
97C22×D3—42no—
98C3×Q8—22no—
99C3×Q8—22no—
100C3×Q8—41no—
101C3×Q8—41no—
102C2×C3×C4—22no—
103C2×C3×C4—41no—
104C2×C3×C4—41no—
105C2×C3×C4—41no—
106C2×C3×C4—42no—
107C2×C3×C4C2×D411yescharacteristic
108C2×C3×C4C2×D411yescharacteristic
109C2×C3×C4C2×D411yescharacteristic
110C2×C3×C4C2×D411yescharacteristic
111C2×C3×C4C2×D411yescharacteristic
112C2×C3×C4#4111yescharacteristic
113C2×Dic3—22no—
114C2×Dic3—22no—
115C2×Dic3—22no—
116C2×Dic3—41no—
117C2×Dic3—41no—
118C2×Dic3SD1614yescomplemented
119D12—22no—
120D12—22no—
121D12—41no—
122D12—41no—
123Dic6—22no—
124Dic6—22no—
125Dic6—41no—
126Dic6—41no—
127C3×C8—22no—
128C3⋊C8—22no—
129C24—61no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8—32no—
136C2×Q8—61no—
137C2×Q8—61no—
138C2×Q8—61no—
139C2×Q8—61no—
140C2×Q8—61no—
141C2×Q8D1212yes—
142C2×D4—32no—
143C2×D4—61no—
144C2×D4—61no—
145C2×D4—61no—
146C2×D4—61no—
147C2×D4—61no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—32no—
151C22×C4—61no—
152C22×C4C22×D311yescharacteristic
153SD16—68no—
154SD16—68no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—32no—
158C2×C8—61no—
159C2×C8—61no—
160C2×C8#6712yes—
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4—68no—
164C22⋊C4—124no—
165C42—61no—
166C42—61no—
167C22×C3#12011yescharacteristic
168C22×C3#12111yescharacteristic
169C22×C3#12711yescharacteristic
170C22×C3C2×SD1612yes—
171C22×C3#13712yes—
172C2×D3—42no—
173C2×D3—44no—
174C2×D3—82no—
175C2×D3—81no—
176C2×D3—81no—
177C3×C4—42no—
178C3×C4—81no—
179C3×C4#12011yescharacteristic
180C3×C4#12011yescharacteristic
181C3×C4#12111yescharacteristic
182C3×C4#12111yescharacteristic
183Dic3—24no—
184Dic3—42no—
185C23—62no—
186C23—121no—
187C23—122no—
188C23—122no—
189C23—121no—
190C23D3×D411yescharacteristic
191Q8—22no—
192Q8—22no—
193Q8—41no—
194Q8—41no—
195Q8—62no—
196Q8—62no—
197Q8—121no—
198Q8—121no—
199D4—62no—
200D4—62no—
201D4—121no—
202D4—121no—
203C2×C4—22no—
204C2×C4—34no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—42no—
208C2×C4—41no—
209C2×C4—62no—
210C2×C4—62no—
211C2×C4—62no—
212C2×C4—121no—
213C2×C4—121no—
214C2×C4C2×D1211yescharacteristic
215C2×C4#23511yescharacteristic
216C2×C4D3×D411yescharacteristic
217C2×C4D3×D411yescharacteristic
218C2×C4D3×D411yescharacteristic
219C2×C4C2×#6711yescharacteristic
220C8—22no—
221C8—62no—
222C2×C3#39211yescharacteristic
223C2×C3#44812yes—
224C2×C3#46112yes—
225C2×C3#49211yescharacteristic
226C2×C3#49711yescharacteristic
227D3—82no—
228C22—122no—
229C22—124no—
230C22—241no—
231C22—241no—
232C22—242no—
233C22#88211yescharacteristic
234C22#88411yescharacteristic
235C22D3×SD1612yes—
236C22#91812yes—
237C22#94011yescharacteristic
238C4—42no—
239C4—64no—
240C4—81no—
241C4—122no—
242C4#88211yescharacteristic
243C4#88411yescharacteristic
244C4#93011yescharacteristic
245C4#93711yescharacteristic
246C3#201311yescharacteristic, complemented, Sylow
247C2—242no—
248C2#480612yes—
249C2#481812yes—
250C2#496011yescharacteristic
251C2#517511yescharacteristic
252C2#519011yescharacteristic
subgroupG/H
centerC23D3×D4
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC2×C3×#103C2
socleC23×C3C2×D4
prime radicalC22×D12C22
hypercenterC2×#103D3
hypocenterC3#2013
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#944C2semidirect
4. C2×C3×#103C2semidirect
6. C2×#810C2semidirect
7. #4550C2semidirect
32. C2×#103D3semidirect
118. C2×Dic3SD16semidirect
246. C3#2013semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#9442
C22×#2152
C22×#2042
C2×C3×#1032
#49882
C2×#8102
#45502
#20133